1 A Hilbert Space Theory of Generalized Graph Signal Processing Feng Ji and Wee Peng Tay, Senior Member, IEEE Abstract Graph signal processing (GSP) has become an important tool in many areas such as image pro- cessing, networking learning and analysis of social network data. In this paper, we propose a broader framework that not only encompasses traditional GSP as a special case, but also includes a hybrid framework of graph and classical signal processing over a continuous domain. Our framework relies extensively on concepts and tools from functional analysis to generalize traditional GSP to graph signals in a separable Hilbert space with infinite dimensions. We develop a concept analogous to Fourier transform for generalized GSP and the theory of filtering and sampling such signals. Index Terms Graph signal proceesing, Hilbert space, generalized graph signals, F-transform, filtering, sampling I. INTRODUCTION Since its emergence, the theory and applications of graph signal processing (GSP) have rapidly developed (see for example, [1]–[10]). Traditional GSP theory is essentially based on a change of orthonormal basis in a finite dimensional vector space. Suppose G = (V; E) is a weighted, arXiv:1904.11655v2 [eess.SP] 16 Sep 2019 undirected graph with V the vertex set of size n and E the set of edges. Recall that a graph signal f assigns a complex number to each vertex, and hence f can be regarded as an element of n C , where C is the set of complex numbers. The heart of the theory is a shift operator AG that is usually defined using a property of the graph. Examples of AG include the adjacency matrix and the Laplacian matrix of G. Graph shift operators are typically chosen to be symmetric. By the This research is supported by the Singapore Ministry of Education Academic Research Fund Tier 2 grant MOE2018-T2-2-019. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore (e-mail: [email protected], [email protected]). 2 n spectral theorem, all the eigenvalues of AG are real and C has an orthonormal basis consisting of eigenvectors of AG. Therefore, one can define the notion of vertex and frequency domains, on which the rest of the theory builds upon. However, instead of assigning a complex number at each vertex, one can assign mathematical objects with richer structures to each vertex of a graph. An example of such a mathematical object comes from a Hilbert space. In particular, it would be of interest to assign an L2 function over a finite closed interval [a; b] to each vertex of a graph. Such a consideration is not just a plain generalization, as it has important practical applications in for example, sensor networks and social networks, where each node in the network is observing a time-varying continuous signal. We have the following considerations. Example 1. (a) The case where each vertex signal is a complex number belongs to the traditional GSP framework, which has been extensively studied in [1]–[10]. (b) An extension of traditional GSP to the case where each vertex signal is a finite-length discrete time series is proposed in the time-vertex GSP framework [8]. Such a vertex signal is from a Hilbert space Cm for some m ≥ 1. The assumption here is that all the time series at different vertices share the same time indices. (c) The case where the signal at each graph vertex v 2 V is itself a graph signal on a finite graph Gv has not been studied in the literature, to the best of our knowledge. In this case, each vertex signal is from Cm for some m ≥ 1, but unlike the time-vertex GSP framework, the underlying graph topology can be more general (the time-vertex GSP framework is equivalent to using a path graph with m vertices). Furthermore, different vertices can have different underlying graphs for their signals, and the correspondence between the vertices in Gv and those in Gu for v 6= u need not be one-to-one. Applications include relating the signals in one social network to another social network, or in problems where the underlying graph at each vertex of G is varying (cf. Section VI-D). For example, when studying signals on time-varying adaptive networks ( [11]–[14]), which includes social networks, biological networks and neural networks, we may let G be a path graph representing a finite portion of the time line. The signal ft at each node t is a vector in Cm. Due to the additional local structures in the signal, it is usually advantageous to consider these structures when performing signal analysis. Thus, we can regard ft as a 3 graph signal corresponding to time t. Furthermore, multiple graph signals ft from different networks can be related to each other, forming yet another underlying graph at each time t. (d) The case where the signal at each vertex is drawn from an infinite dimensional Hilbert space has again never been studied. An example is continuous time signals that are not bandlimited (in the time direction). In this case, from the Shannon-Nyquist Theorem [15], it is impossible to recover the full undistorted signal using a finite sampling rate (cf. Section VI-B). Thus, the time-vertex GSP framework, which requires discrete time series at all graph vertice, would introduce errors in the inference procedure. Furthermore, signals may not be sampled synchronously in a sensor network (sensor synchronization usually requires additional effort [16]–[18]) so that the time-vertex GSP framework may not be the best approach. A more general GSP framework is thus needed to address many practical engineering scenarios. In addition, a Hilbert space, such as the space of L2 functions over an appropriate domain Ω, usually has rich internal structures. The usual GSP framework takes a “snapshot picture" by looking at the graph signal at x 2 Ω one by one. This approach may easily disregard the internal relations among the points in Ω. In this paper, the proposed framework avoids such a local consideration entirely, and fuses the graph operators with operations on L2(Ω) for signal processing to achieve minimal information loss. For example, to study continuous time graph signals mentioned above, it can be more beneficial to consider the graph signal to be a function belonging to L2(R), such that we may apply operations such as continuous Fourier transform and wavelet transforms in conjunction with graph based transforms. We shall demonstrate this by the example of information propagation on social networks in Section VI-B. In this paper, we propose and develop a Hilbert space theory of generalized GSP that can handle all the above cases. To do that, it is inevitable that the theory is based on an abstract foundation, which requires the reader to have a good grasp of functional analysis concepts and tools. For these, we refer the reader to [19], [20]. We shall constantly refer back to Example 1 to illustrate and explain theoretical results. We shall develop the theory parallel to classical signal processing and GSP. Important notions such as convolution filters and bandlimitedness are best understood when signals are viewed in the frequency domain. Therefore, defining what constitutes the frequency domain is a hallmark of both classical signal processing and traditional GSP. In the same spirit, we first define the 4 frequency domain for the generalized GSP framework using the spectral theory of compact operators on Hilbert spaces. We then proceed to discuss filtering and sampling. A preliminary version of this paper was presented in the conference paper [21], which introduced some of the basic concepts in this paper without proof. The rest of the paper is organized as follows. In Section II, we set up the framework by defining graph signals in a separable Hilbert space. In Section III, we introduce the notion of frequency. Based on this, we develop the concepts of filtering and sampling in Section IV and Section V. We present numerical results in Section VI and conclude in Section VII. Throughout, we provide examples to highlight situations where the framework of the paper is not only applicable but also necessary. Notations. We use R and C to denote the real and complex fields, respectively, while Z denotes the set of integers. The symbol ⊗ denotes tensor product, ◦ is function composition, and ∼= p denotes isomorphic equivalence. Id is the identity operator and i = −1. Cn is equipped with 2 2 an inner product h·; ·i n with corresponding norm k·k n . The space L (Ω) = L (Ω; F; µ) with C C R 2 (Ω; F; µ) a measure space, is the collection of functions f :Ω 7! C such that Ω jfj dµ < 1. II. GENERALIZED GRAPH SIGNALS Let G = (V; E) be a simple finite undirected weighted graph and X be a metric space. In this paper, when we talk about a vector space, we always assume that the base field is the complex numbers C, unless otherwise stated. In traditional GSP, the signals on the vertices of G are assumed to be real or complex. We generalize this as follows. Definition 1. Suppose H is a Hilbert space (i.e., a complete inner product space). A graph signal in H is a function f : V !H: The space of graph signals in H is denoted by S(G; H): With a few examples below, we demonstrate why this generalized notion is versatile. In short, Hilbert spaces include both the more familiar finite dimensional cases and also infinite dimensional cases that can represent more realistic signals in practice.